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G. [W1, Appendix B]. For nonk,p k,p compact base manifolds X we denote by Aloc (X) and Gloc (X) the spaces of sections and maps for which the regularity holds on all compact subsets of X. Next, we describe the class of 4-manifolds that we will be considering. Here and throughout all Riemann surfaces are closed oriented 2-dimensional manifolds. Moreover, unless otherwise mentioned, all manifolds are allowed to have a smooth boundary. e. int X := X \ cl(X \ X ) might intersect ∂X. 48 K. 1. A 4-manifold with a boundary space-time splitting is a pair (X, τ ) with the following properties: (i) X is an oriented 4-manifold (with boundary) which can be exhausted by a nested sequence X= Xk , k∈N where all Xk are compact submanifolds and deformation retracts of X such that Xk ⊂ int Xk+1 for all k ∈ N.
Note that the scalar-flatness of g¯ Vλ implies c1 (P1 × P1 )·[ Vλ ] = 0. Furthermore, it follows from (10) that [ Vλ ]·[ωpt×S 2 ] = [ 0 ]2 , where 0 is the K¨ahler form of the standard product metric g0 = −hP1 ⊕ hP1 . Then the cohomology class [ Vλ ] is independent of λ. By Moser’s theorem [16], we see that (M, ¯ V ) is symplectomorphic to (S 2 × S 2 , 0 ). Concerning the Weyl conformal tensor W of g¯ V , we first note the following proposition, which is verified by a direct computation. Proposition 8.
The case 2 < p ≤ 4, when W 1,p -functions are not automatically continuous, poses some special difficulties in this last step. Firstly, in order to obtain regularity results from the Cauchy-Riemann equation, one has to straighten out the Lagrangian submanifold by going to suitable coordinates. This requires a C 0 -convergence of the connections, which in case p > 4 is given by a standard Sobolev embedding. In case p > 2 one still obtains a special compact embedding W 1,p (U × ) → C 0 (U, Lp ( )) that suits our purposes.